It's true villain's play look rather strong, but another fact you should consider is the fact this hand is actually BB defending vs CO steal and can thus be looser pre and postflop.

On the flop I believe folding would be too weak here against the minraise because if we do so then it makes sense for villain to check/minraise any hand and another thing is I am sure he can do with quite wide range such as worse Axs/FDs/2-pairs/sets/straights.

On the turn, we can call against his half pot bet and then I would re-evaluate river and usually fold to a river shove, specially if a
shows up.